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System-Level Analysis and Optimization of PressureRetarded Osmosis for Power Generation
Mingheng Li
Department of Chemical and Materials Engineering
California State Polytechnic University, Pomona
Nov. 12, 2015
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OUTLINE
1 Motivation
2 Simple Analysis in Literature
3 Analysis and Optimization Accounting for Flux ProfileBasic definitionswithout Concentration Polarizationwith Concentration Polarization
4 Concluding Remarks
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MOTIVATION OF THIS WORK
Pressure retarded osmosis is one of the promising “green” energytechniques that produces power from salinity gradient.
There is a lack of computational framework to address scale-up issuesin PRO
◮ Optimal applied pressure much lower than ∆π/2 is observed in someexperiments (Xu et al., JMS, 2009; She et al., JMS, 2012; Sharif et al.,Membranes, 2014).
◮ Power generation in PRO is much lower than theoretical prediction.
Mathematical models helps elucidate the understanding of energyissues in PRO.
◮ Power density
◮ Specific Energy Production (SEP)
◮ Efficiency of osmotic to hydraulic energy
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SIMPLE ANALYSIS BASED ON CONSTANT FLUX
(see, e.g. Review by Helfer et al. JMS, 2014)
π0D
Q0
)
0
qdQ0
Q0(1−q
d
PRO
PUMP
TURBINE
FS
DS
πF
Constant water flux along membrane✞✝ ☎✆Jw = dQ/dA = Lp(∆π −∆P )
Power Density (PD)✞✝ ☎✆PD = Jw∆P = Lp(∆π −∆P )∆P✞✝ ☎✆∆Popt = ∆π/2
This may be true for short membranes (i.e. under lab-scaleexperimental conditions)
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BASIC DEFINITIONS IN PRO
π0D
Q0
)
0
qdQ0
Q0(1−q
d
PRO
PUMP
TURBINE
FS
DS
πF
Specific Energy Production (SEP)✞✝ ☎✆SEP = Q0(qd − 1)∆P/Q0 = (qd − 1)∆P
qd: dilution ratio at the end of the membrane∆P : applied pressure
Normalized SEP or Osmotic to Hydraulic Efficiency✞✝ ☎✆NSEP = SEP/πD0
= (Q0SEP )/(Q0πD0) = ηO2H
Power Density (PD)✞✝ ☎✆PD = (NSEP )(Q0/A)πD0
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1D MATHEMATICAL MODEL WITHOUT CP
Local water flux (assuming πF = πF0)✎
✍☞✌
dQ
dA= Lp(∆π −∆P ) = Lp(π
D0
Q0
Q− πF
0 −∆P )
Dimensionless form ✎✍
☞✌
dq
dx= γ
(
1
q− θ
)
✞✝ ☎✆q = Q/Q0, θ = (∆P + πF0)/πD
0, γ = ALpπ
D0/Q0
Solution (Li, AIChE J., 2015)✓✒
✏✑γ =
1
θ
[
1− qd +1
θln
1− θ
1− qdθ
]
qd: dilution ratio at the end of the membrane
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SOLUTION TO qd AND NSEP
Profiles of qd and NSEP
0
5
10
0
0.5
11
2
3
4
5
γθ
q d
0
5
10
0
0.5
10
0.2
0.4
0.6
0.8
γθ
NS
EP
Observations
◮ qd increases as ∆P reduces and/or γ increases.
◮ At a fixed γ, there is an optimal ∆P corresponding the maximumNSEP.
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CONSTRAINED NONLINEAR OPTIMIZATION
Optimization model to determine optimal NSEP and ∆P✬
✫
✩
✪
maxα,z
NSEP = (qd − 1)
(
1
α− r
)
s.t.qd = α− (α− 1)e−z
γ = α(1− qd + αz)
1− α ≤ 0
1− qd ≤ 0
Optimization of PD and optimization NSEP are essentially equivalentif Q0/A and πD
0are given, since PD = (NSEP )(Q0/A)π
D0.
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CONSTRAINED NONLINEAR OPTIMIZATION OF NSEP
Optimization results
10−1
100
101
102
0.1
0.2
0.3
0.4
0.5
γ
(∆P
/∆π 0) op
t
r = 0r = 0.1r = 0.2r = 0.4
10−1
100
101
102
1
2
3
4
5
6
7
γ
q d opt
r = 0r = 0.1r = 0.2r = 0.4
10−1
100
101
102
0
0.2
0.4
0.6
0.8
1
γ
NS
EP
opt
r = 0r = 0.1r = 0.2r = 0.4
Observations
◮ The optimal ∆P shifts away from ∆π0/2 as γ = ALpπD0/Q0 increases.
◮ An increase in r (r = πF0/πD
0) significantly reduces qd and NSEP.
◮ When r = 0 (or fresh water is used as feed solution), the largest NSEPoccurs at forward osmosis conditions, i.e., ∆Popt = 0.
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OPTIMAL DRIVING FORCE IN PRO
Optimal dimensionless driving force ζ = (∆π −∆P )/πD0
10−1
100
101
102
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
γ
ζ opt
InletOutletAverage
10−1
100
101
102
0
0.1
0.2
0.3
0.4
0.5
γ
ζ−
opt
r = 0r = 0.1r = 0.2r = 0.4
Inlet Outlet Average (ζ)
1− θ 1/qd − θ (qd − 1)/γ
Similarity between RO and PRO◮ A larger γ allows the PRO to be operated closer to its thermodynamic
limit, thus improving SEP.
◮ A larger γ allows the RO to be operated closer to its thermodynamiclimit, thus improving SEC (Li, IECR, 2010; Li, Desalination, 2012, Li,IECR, 2013).
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NONLINEARITIES IN PRO
Profiles of power efficiency and density when FS is fresh water
A
η O2H
Lp
η O2H
∆π0
η O2H
Q0
η O2H
A
PD
Lp
PD
∆π0
PD
Q0
PD
Power density reduces as membrane areas increases. Therefore, itsvalue in plant operation might be smaller than the one measuredunder lab-scale experimental conditions even if the same type ofmembrane is used.
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PRO MODEL ACCOUNTING FOR CP
Water flux accounting for concentration polarization (Achilli et al.,JMS, 2009, Xu et al., JMS, 2010)★
✧
✥
✦Jw = Lp
πDb exp(− Jw
km)1−
πF
b
πD
b
exp(JwK) exp( Jwkm
)
1 + BJw
[exp(JwK)− 1]−∆P
dQ
dA= Jw
Power Density ✓✒
✏✑PD =
∆P
∫ A
0
JwdA
A
Jw is in implicit form. Intensive computation is required to directlysolve optimal ∆P .
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FLUX IN EXPLICIT FORM UNDER TWO
ASSUMPTIONS
If Jw/km << 1 (Achilli et al., JMS, 2009) and JwK << 1 (Lee etal., JMS, 1981), Jw may be approximated by (Li, AIChE J., 2015)☛
✡✟✠Jw ≈ L′
p(σ∆π −∆P )
where σ = 1/(1 +BK) and L′
p = Lp/(1 + Lp∆πσ/km).
Under such assumptions an analytical solution in PRO may beobtained (Li, AIChE J., 2015)✓✒
✏✑γ =
1
θ
[
(1− qd) +σ
θln
σ − θ
σ − qdθ
]
where γ = AL′
pQ0/πD0.
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SHORT-CUT OPTIMIZATION ACCOUNTING FOR CP
Optimization model✬
✫
✩
✪
maxα,z
NSEP = (qd − 1)
(
1
α− r
)
s.t.qd = ασ − (ασ − 1)e−z
γ = α(1− qd + ασz)
1/σ − α ≤ 0
1− qd ≤ 0
It is found that the model provides very accurate solution to ∆Popt.However, flux profile, NSEP and PD may be better calculated usingthe original concentration polarization model and the derived ∆Popt.
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EFFECT OF ICP IN PRO
10−1
100
101
102
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
γ
θ opt
σ=0.9σ=1
10−1
100
101
102
1
2
3
4
5
6
7
γ
q d opt
σ=0.9σ=1
10−1
100
101
102
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
γ
NS
EP
opt
σ=0.9σ=1
10−1
100
101
102
0
0.1
0.2
0.3
0.4
0.5
γ
ζ−
opt
σ=0.9σ=1
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CASE STUDIES
Parameters used in the studies (Achilli et al., JMS, 2009)
Parameters Value
km 8.48×10−5 m/sK 4.51×105 s/mB 1.11×10−7 m/sLp 1.87×10−9 m/s/kPa∆π0 2763, 4882 kPa
Four different Q0/A (1, 2, 4, 8 ×10−6 m/s) are considered. Thecorresponding γ are between 0.5-10. These are comparable to thereverse process of industrial SWRO (Li, IECR, 2013).
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COMPARISON BETWEEN SHORT-CUT AND
RIGOROUS OPTIMIZATION METHODS
Values obtained using the short-cut optimization method, if different fromthe rigorous method, are presented in parenthesis.
∆π0 = 2763 kPa ∆P , kPa Jw , µm/s PD, W/m2 γ qd ηO2H , %
∆π = ∆π0 1330 (1316) 2.16 (2.18) 2.87 - - -Q0/A = 8µm/s 1300 (1295) 1.78 (1.79) 2.31 0.65 1.22 10.5Q0/A = 4µm/s 1260 (1259) 1.56 (1.56) 1.96 1.29 1.39 17.7Q0/A = 2µm/s 1190 (1186) 1.28 (1.28) 1.52 2.58 1.64 27.6Q0/A = 1µm/s 1070 (1067) 1.00 (1.01) 1.07 5.17 2.01 38.9
∆π0 = 4882 kPa ∆P , kPa Jw , µm/s PD, W/m2 γ qd ηO2H , %
∆π = ∆π0 2390 (2325) 3.38 (3.47) 8.07 - - -Q0/A = 8µm/s 2260 (2244) 2.64 (2.66) 5.97 1.14 1.33 15.3Q0/A = 4µm/s 2140 (2133) 2.24 (2.25) 4.79 2.28 1.56 24.6Q0/A = 2µm/s 1950 (1943) 1.79 (1.80) 3.50 4.56 1.90 35.8Q0/A = 1µm/s 1680 (1685) 1.38 (1.38) 2.32 9.13 2.37 47.4
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EFFECT OF DILUTION AND CP IN PRO
800 900 1000 1100 1200 1300 1400 1500 16000.5
1
1.5
2
2.5
3
3.5
4 ← ∆π
0/2
∆P (kPa)
PD
(W
/m2 )
rigorousshort−cut
500 1000 1500 2000 2500 30000
2
4
6
8
10
12 ← ∆π
0/2
∆P (kPa)
PD
(W
/m2 )
rigorousshort−cut
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3x 10
−6
Fraction of Membrane
J w (
m/s
)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
−6
Fraction of Membrane
J w (
m/s
)
Black: ∆π = ∆π0 and no concentration polarization.Red: ∆π = ∆π0 and with concentration polarization.Magenta: Q0/A = 8× 10−6 m/s. Yellow: Q0/A = 4× 10−6 m/s.Blue: Q0/A = 2× 10−6 m/s. Green: Q0/A = 1× 10−6 m/s.
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CONCLUSIONS
Characteristic equation in PRO to relate membrane properties,operating conditions and performance✓
✒✏✑γ =
1
θ
[
(1− qd) +σ
θln
σ − θ
σ − qdθ
]
where γ = AL′
pQ0/πD0, θ = (∆P + πF
0)/πD
0, σ = 1/(1 +BK).
Short-cut optimization yields essentially the same solution as therigorous solution. Moreover, it provides parameters to explain theeffect of ICP, ECP and dilution in DS.
Shift of optimal ∆P from ∆π0/2.
◮ Dilution effect (i.e. γ is not zero).
◮ Internal concentration polarization (i.e., σ < 1).
Nonlinearities and conflicting power density and efficiency in processscale-up.
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ACKNOWLEDGEMENT AND RELATED PUBLICATIONS
Financial Support from American Chemical Society Petroleum ResearchFund (No. 55347-UR9) is gratefully acknowledged.
Li, M. “Analysis and Optimization of Pressure Retarded Osmosis for PowerGeneration, ” AIChE Journal, 61, 1233-1241, 2015.Li, M. “Energy Consumption in Spiral Wound Seawater Reverse Osmosis at theThermodynamic Limit,” Ind. Eng. Chem. Res., 53, 3293-3299, 2014.Li, M. “A Unified Model-Based Analysis and Optimization of Specific EnergyConsumption in BWRO and SWRO,” Ind. Eng. Chem. Res., 52, 17241-17248,2013.Li, M.; Noh, B. “Validation of Model-Based Optimization of Reverse Osmosis(RO) Plant Operation,” Desalination, 304, 20-24, 2012.Li, M. “Optimization of Multitrain Brackish Water Reverse Osmosis (BWRO)Desalination,” Ind. Eng. Chem. Res., 51, 3732-3739, 2012.Li, M. “Optimal Plant Operation of Brackish Water Reverse Osmosis WaterDesalination,” Desalination, 293, 61-68, 2012.Li, M. “Reducing Specific Energy Consumption in Reverse Osmosis WaterDesalination: An Analysis from First Principles,”Desalination, 276, 128-135, 2011.Li, M. “Minimization of Energy in Reverse Osmosis Water Desalination usingConstrained Nonlinear Optimization,” Ind. Eng. Chem. Res., 49, 1822-1831, 2010.
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